Suppose a firm has the following Leontief production function: q = min{L, 3K} a. What is the optimal ratio of workers to capital? Are L and K perfect input complements or perfect input substitutes? b. What is the least cost combination of L and K that the firm should employ to produce 90 units of output when w/r = 1. Does this optimal combination of L and K change when w/r changes? Explain your answer briefly. c. More generally, solve for the firm’s factor demand functions for L and K. d. Using these factor demand functions, derive the firm’s long run cost function. e. What is the marginal cost of an additional unit of q? What is the average cost of each unit of q? How do marginal cost and average cost vary with q?

Detailed step-wise answer is given below:

Explanation:

q = min{L, 3K}

(a)

This is a fixed proportion production function which signifies that L and K are perfect complements. In this case, cost is minimized when

L = 3K, or

L/K = 3

(b)

When q = 90, we have

90 = min{L, 3K}

90 = min{L, L}

L = 90

K = L/3 = 90/3 = 30

The (L/K) ratio does not change with a change in (w/r), since the inputs are used in fixed proportion.

(c)

Generalized isocost line: C = wL + rK

C = wL + r x (L/3)

3C = 3wL + rL

3C = L x (3w + r)

L = 3C / (3w + r) [Generalized factor demand function for L]

K = [3C / (3w + r)] / 3 = C / (3w + r) [Generalized factor demand function for K]

(d)

Since q = min{L, 3K} and L = 3K,

q = min{L, L} = L

K = L/3 = q/3

Total cost (C) = wL + rK = wq + r x (q/3)

(e)

Marginal cost (MC) = dC/dq = w + (r/3)

Average cost (MC) = C/q = w + (r/3)

Therefore, MC and AC are independent of q.

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